52 research outputs found
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Perfect matchings in random sparsifications of Dirac hypergraphs
For all integers , let be the minimum
integer such that every -uniform -vertex hypergraph with minimum -degree at least has an optimal
matching. For every fixed integer , we show that for and , if is an -vertex
-uniform hypergraph with , then
a.a.s.\ its -random subhypergraph contains a perfect matching
( was determined by R\"{o}dl, Ruci\'nski, and Szemer\'edi for all
large ). Moreover, for every fixed integer and
, we show that the same conclusion holds if is an
-vertex -uniform hypergraph with . Both of these results strengthen Johansson, Kahn,
and Vu's seminal solution to Shamir's problem and can be viewed as "robust"
versions of hypergraph Dirac-type results. In addition, we also show that in
both cases above, has at least many perfect matchings, which is best possible up to a
factor.Comment: 25 pages + 2 page appendix; Theorem 1.5 was proved in independent
work of Pham, Sah, Sawhney, and Simkin (arxiv:2210.03064
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
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